Calculus, Gauge Theory and Noncommutative Worlds

TL;DR

This paper develops a non-commutative calculus framework where gauge structures, connections, and metric variability naturally emerge, unifying aspects of gauge theory, Hamiltonian, and quantum mechanics in a novel mathematical setting.

Contribution

It introduces a non-commutative calculus that naturally incorporates gauge theoretic structures and generalizes the Levi-Civita connection, extending Weyl's ideas on metric variability.

Findings

Gauge structures arise naturally in non-commutative calculus

A covariant Levi-Civita connection is derived in this framework

The metric exhibits wider variability than classical metrics

Abstract

This paper shows how gauge theoretic structures arise naturally in a non-commutative calculus. Aspects of gauge theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative framework for calculus and differential geometry. We show how a covariant version of the Levi-Civita connection arises naturally in this commutator calculus. This connection satisfies the formula $$\Gamma_{kij} + \Gamma_{ikj} = \nabla_{j}g_{ik} = \partial_{j} g_{ik} + [g_{ik}, A_j].$$ and so is exactly a generalization of the connection defined by Hermann Weyl in his original gauge theory. In the non-commutative world $\cal N$ the metric indeed has a wider variability than the classical metric and its angular holonomy. Weyl's idea was to work with such a wider variability of the metric. The present formalism provides a new context for Weyl's original idea.